How to create clickable register/login icons in top right header

I am working with the Modality wordpress theme and trying to re-create the top right header section from the following wordpress theme:

https://crunchpress.com/demo/prayer/

After testing, coding, breaking the header.php, and hunting for almost every plugin, I can’t get the social icons, search, sign up, and sign in in that position and to get them to work.

Any ideas?

Thanks.